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Categories of Continuous Functors

Posted by Emily Riehl

as part of the Kan Extension Seminar series of lectures. I warmly thank all the participants and the organizer Emily Riehl for giving me this way of escape to the woeful solitude a “baby” category theorist (like I am preparing to become) suffers here in Italy. It is an amazing and overwhelming experience, I can’t even estimate the amount of things I already learned after these three lectures. There are two other people without whom this wouldn’t have been possible: my current advisor, D. Fiorenza, who patiently helped me to polish the exposition you are about to read, and my friend Paolo, for which the words “I don’t want to learn this” are meaningless.

That said, let’s begin with the real discussion.

Freyd and Kelly’s paper was the first to raise and solve in a very elegant way some fundamental questions in elementary Category Theory, the so-called Orthogonal subcategory problem, and Continuous functor problem.

Orthogonality between arrows

Definition. An object BB in a category A\mathbf{A} is said to be orthogonal to an arrow k:A→Xk\colon A\to X (we say k⊥Bk\perp B) if the hom-functor A(−,B)\mathbf{A}(-,B) sends kk to a bijection A(X,B)→A(A,B)\mathbf{A}(X,B)\to \mathbf{A}(A,B) between sets.

If A\mathbf{A} has a terminal object, then k⊥Bk\perp B if and only if the terminal arrow B→1B\to 1 has the so-called (unique) right lifting property against kk: this means that for any choice of ff in the diagram
A⟶fBk↓↓X⟶1
\begin{array}{ccc} A &\stackrel{f}{\longrightarrow}& B \\ {}^k \downarrow && \downarrow \\ X & \longrightarrow & 1 \end{array}
there exists a unique arrow a:X→Ba\colon X\to B making the upper triangle commute. Obviously, there is a dual notion of left lifting property.

Statement of the problem: the CFP and the OSP.

Orthogonal Subcategory Problem (OSP). Given a class ℋ\mathcal{H} of arrows in a category A\mathbf{A}, when is the full subcategory ℋ⊥{\mathcal{H}}^\perp of all objects orthogonal to ℋ\mathcal{H} a reflective subcategory (i.e., when there exists a left adjoint to the inclusion ℋ⊥↪A{\mathcal{H}}^\perp\hookrightarrow \mathbf{A})? ▪\blacksquare

There are lots of “protoypical” examples of the OSP in Algebra and Geometry: think for example to the case of sheaves of sets (on a given Grothendieck site) as a reflective subcategory among presheaves: the sheaf condition can be easily stated in terms of an orthogonality request: a presheaf FF on a site (C,J)(\mathbf{C},J) is a JJ-sheaf if and only if
iC⊥F∀C∈Ci_C\perp F \quad \forall C\in\mathbf{C}
for every covering sieveiC:S→yC(C)=C(−,C)i_C\colon S\to y_{\mathbf{C}}(C)=\mathbf{C}(-,C).

In fact this is not a case, since following a general tenet “(at least some) localizations are determined by an orthogonality class” (see for example the definition of 𝒮\mathcal{S}-local object and the nnLab page about the OSP).

Freyd and Kelly were the first to point out that a solution for the OSP turns out to solve another fundamental question, which falls under the name of “continuous functor problem”:

Continuous Functor Problem (CFP). Given a category C\mathbf{C} and a class of diagrams (say Γ\Gamma) in it, when is the category of all functors C→D\mathbf{C}\to \mathbf{D} which preserve limits of all Γ\Gamma-shaped diagrams reflective in the category of functors C→D\mathbf{C}\to \mathbf{D}? ▪\blacksquare

(Important) Remark. Before going on, we must spend a word on the notion of continuity: Freyd and Kelly published an erratum shortly after the paper, to correct the “stupid mistake of supposing that the limit of a constant diagram is the constant itself”; counterexamples to this statement abound, and in fact it can be easily shown that the limit of the constant functorΔ(C):J→C\Delta(C)\colon \mathbf{J}\to\mathbf{C} is (whenever this copower exists) precisely Cπ0(J)C^{\pi_0(\mathbf{J})}, where π0(J)\pi_0(\mathbf{J}) is the set of connected components of the category J\mathbf{J}.

Once this is fixed, notice that the CFP arises in an extremely elementary way: for example,

An additive functor FF between abelian categories is left exact if and only if it commutes with finite limits, and

The above sheaf condition can be easily restated in the good old familiar continuity request on coverings of objects C∈CC\in\mathbf{C}.

This should give you evidence that the two problems are not unrelated:

Proposition. Given a class of diagrams Γ\Gamma in a small complete category C\mathbf{C}, we get a family of natural transformations
𝒢(Γ)={mγ:colimC(γ,−)→C(limγ,−)}γ∈Γ
\mathcal{G}(\Gamma)=\Big\{ m_\gamma \colon \colim \mathbf{C}(\gamma,-) \to \mathbf{C}\big( \lim\; \gamma,- \big) \Big\}_{\gamma\in\Gamma}
and a functor F:C→SetF\colon \mathbf{C}\to Set is Γ\Gamma-continuous if and only if it is orthogonal to each arrow in 𝒢(Γ)\mathcal{G}(\Gamma).

OSP ⇒\Rightarrow CFP: Strategy of the proof.

The strategy adopted by Freyd and Kelly to solve the OSP, is to find sufficient conditions on ℋ\mathcal{H} so that Freyd’s Adjoint Functor Theorem applies to the inclusion ℋ⊥↪A\mathcal{H}^\perp\hookrightarrow \mathbf{A} (in particular, since it can be shown that ℋ⊥\mathcal{H}^\perp is always complete, this boils down to find a solution set for ℋ⊥\mathcal{H}^\perp to apply Freyd Adjoint Functor Theorem).

These conditions are of 1+3 different types:

Cocompleteness;

The presence of a proper factorization system;

The presence of a generator;

A (global) boundedness condition (or equivalently, on the generator in the previous point).

Factorization systems

Notation. We denote llp(ℋ)llp(\mathcal{H}) (resp, rlp(ℋ)rlp(\mathcal{H})) the (possibly large) class of all arrows left (resp, right) orthogonal to each arrow of the class ℋ\mathcal{H}.

Definition. A prefactorization system on a category A\mathbf{A} consists of two classes of arrows 𝔽=(ℰ,ℳ)\mathbb{F}=(\mathcal{E},\mathcal{M}) such that ℰ=llp(ℳ)\mathcal{E} = llp(\mathcal{M}) and ℳ=rlp(ℰ)\mathcal{M} = rlp(\mathcal{E}).

A prefactorization system 𝔽\mathbb{F} on A\mathbf{A} is said proper if ℰ⊂Epi\mathcal{E}\subset Epi and ℳ⊂Mono\mathcal{M}\subset Mono.

A factorization system (OFS, or simply FS) on a category A\mathbf{A} corresponds to the modern notion of orthogonal factorization system: a (proper) factorization on the category A\mathbf{A} is precisely a (proper) prefactorization 𝔽=(ℰ,ℳ)\mathbb{F}=(\mathcal{E},\mathcal{M}) such that each f:X→Yf\colon X\to Y can be written as a composition X→eW→mYX\stackrel{e}{\to}W\stackrel{m}{\to}Y with e∈ℰ,m∈ℳe\in \mathcal{E}, m\in\mathcal{M}.

Examples.
0. Any category C\mathbf{C} has two trivial factorization systems, namely (MorC,IsoC)( Mor_\mathbf{C} , Iso_\mathbf{C} ) and (IsoC,MorC)( Iso_\mathbf{C} , Mor_\mathbf{C} ), where IsoCIso_\mathbf{C} denotes the class of all isomorphisms, and MorCMor_\mathbf{C} the class of all arrows in C\mathbf{C};
1. The category SetSet has a factorization system 𝔽=(Epi,Mono)\mathbb{F}=(Epi,Mono) where EpiEpi denotes the class of surjective maps, and MonoMono the class of injective maps. More generally, the category of models of any algebraic theory (monoids, (abelian) groups, …) has a proper FS (Epi*,Mono)(Epi^\ast, Mono), where Epi*Epi^\ast is the class of extremal epimorphisms (which may or may not coincide with plain epimorphisms); and for abelian categories, (elementary) toposes…

Generators

Definition. If A\mathbf{A} is a category with a proper factorization system 𝔽\mathbb{F}, we say that a family of objects {qi:Bi→C}i∈I\{q_i\colon B_i\to C\}_{i\in I}lies in ℰ\mathcal{E} if there exists a uniquet:C→Xt\colon C\to X solving (all at once) the lifting problems
Bi⟶fiXqi↓↓mC⟶Y
\begin{array}{ccc}
B_i & \stackrel{f_i}{\longrightarrow} & X \\
{}^{q_i} \downarrow && \downarrow^m\\
C & \longrightarrow & Y
\end{array}
(one for each i∈Ii\in I). If A\mathbf{A} has sufficiently large coproducts, this condition is obviously equivalent to ask that the arrow (q¯:⨿i∈IBi→C)∈ℰ\left(\bar q\colon \amalg_{i\in I} B_i\to C\right)\in\mathcal{E}.

Definition. A generator in a category with a proper factorization system 𝔽=(ℰ,ℳ)\mathbb{F}=(\mathcal{E}, \mathcal{M}) consists of a small full subcategory G⊆A\mathbf{G}\subseteq\mathbf{A} such that for any A∈AA\in\mathbf{A} the family {G→A}G∈G\{G\to A\}_{G\in\mathbf{G}} lies in ℰ\mathcal{E} in the former sense.

Remark. Mild completeness assumptions on A\mathbf{A} entail that

A generator separates objects, i.e. if f≠gf\neq g then there exists an object G∈GG\in\mathbf{G} and an arrow G→AG\to A such that fk≠gkf k\neq g k.

For extremal FSs (in which the left/right class coincides with that of extremal epi/mono) the converse of 1,2 is also true, so as to recover the notion of generator as a “separator for objects”.

Boundedness

Notation. In this section A\mathbf{A} admits all limits and colimits whenever needed

Definition(s). An ordered set JJ is said to be σ\sigma-directed (for a regular cardinalσ\sigma) if every subset of JJ with less than σ\sigma elements has an upper bound in JJ. A σ\sigma-directed family {Cj→B}j∈J\{C_j\to B\}_{j\in J} of subobjects of B∈AB\in\mathbf{A} consists of a functor J→SubA(B)J\to Sub_\mathbf{A}(B) from a σ\sigma-directed set to the posetal class of subobjects of BB. The colimit of such a functor, denoted ⋃j∈JCj\bigcup_{j\in J} C_j is called the σ\sigma-directed union of the family.

With these conventions, we say that an object A∈AA\in\mathbf{A} is bounded by a regular cardinal σ\sigma (called the bound of AA) if every arrow A→⋃j∈JCjA\to \bigcup_{j\in J} C_j to a σ\sigma-directed union factors through one of the CjC_j. The category A\mathbf{A} is bounded if each A∈AA\in\mathbf{A} is bounded by a regular cardinal σA\sigma_A (possibly depending on AA).

Example. In A=Set\mathbf{A}= Set a set of cardinality ≤σ\le \sigma is σ\sigma-bounded.

Remark.σ\sigma-boundedness is obviously linked to local σ\sigma-presentability: [PK]’s locallyσ\sigma-presentable categories are precisely those categories A\mathbf{A} which

admit arbitrary colimits;

admit a generator G\mathbf{G} each of which object is σ\sigma-presentable.

Examples. Examples of such structures/properties on categories abound:

Any abelian, AB(5)AB(5), bicomplete and bi-well-powered category A\mathbf{A}, is bounded;

Given a regular cardinal σ\sigma, locally σ\sigma-presentable categories are σ\sigma-bounded, and admit a generator with respect to the proper FS (Epi*,Mono)(Epi^\ast, Mono): sets, small categories, presheaf toposes and Grothendieck abelian categories all fall under this example. Less obviously, the converse implications is false: exhibiting a σ\sigma-bounded category with a generator which is not locally σ\sigma-presentable requires to accept the inexistence of measurable cardinals (see [FK], Example 5.2.3).

Solution of the OSP

Theorem (OS theorem). If A\mathbf{A} is complete, cocomplete, bounded and co-well-powered with a proper FS 𝔽=(ℰ,ℳ)\mathbb{F}=(\mathcal{E},\mathcal{M}), and ℋ
\mathcal{H} is a class of arrows whose elements are “almost all” in ℰ\mathcal{E}, i.e. ℋ=𝒮∪ℰ¯\mathcal{H}=\mathcal{S}\cup \overline{\mathcal{E}} (we call these classes quasi-small with respect to ℰ\mathcal{E}), where

𝒮\mathcal{S} is a set;

ℰ¯\overline{\mathcal{E}} is possibly large but contained in ℰ\mathcal{E}.

Then ℋ⊥\mathcal{H}^\perp is a reflective subcategory. ▪\blacksquare

Proof. [FK] performs a clever transfinite induction to generate a solution set for any object A∈AA\in\mathbf{A}: if k:M→Nk\colon M\to N is the typical arrow in 𝒮\mathcal{S}, we define

This is where boundedness comes into play: if σ\sigma is the cardinal bounding AA, then the induction stops at σ\sigma: Sσ,A∩ℋ⊥\S_{\sigma, A}\cap \mathcal{H}^\perp is the desired solution set for A∈AA\in\mathbf{A}, namely every arrow f:A→Bf\colon A\to B whose codomain lies in ℋ⊥\mathcal{H}^\perp factors through some X∈Sσ,A∩ℋ⊥X\in \S_{\sigma, A}\cap \mathcal{H}^\perp.

Solution of the CFP

The procedure we adopted to reduce the CFP to the OSP (building 𝒢(Γ)\mathcal{ G}(\Gamma)) doesn’t take care of any size issue: to repair this deficiency we exploit the following

The particular shape of ℋ\mathcal{H} is due to the procedure used in [FK] to reduce the CFP to the OSP. The ⊗\otimes operation is a copower, in the obvious sense: given β:C→Set\beta\colon \mathbf{C}\to Set, β⊗A:F⊗A→G⊗A\beta\otimes A\colon F\otimes A\to G\otimes A, where F⊗A:C↦FC⊗A=∐c∈FCAF\otimes A\colon C\mapsto F C\otimes A = \coprod_{c\in F C}A.

The key point of this result is that the class ℋ1\mathcal{H}_1 is small (obviously) whenever Θ\Theta is, so we can conclude applying the OS theorem:

Theorem (CF theorem).
Let C\mathbf{C} be a small category, and D\mathbf{D} a bicomplete, bounded, co-wellpowered category with a generator and a proper factorization 𝔽=(ℰ,ℳ)\mathbb{F}=(\mathcal{E},\mathcal{M}). Let Γ\Gamma be a class of cylinders whose elements are almost all cones (this means that the collection of diagrams which are not cones is a set). Then the subcategory of Γ\Gamma-continuous functors is reflective in Fun(C,D)Fun(\mathbf{C},\mathbf{D}). ▪\blacksquare

The rough idea behind this result is the following: ℋ⊥\mathcal{H}^\perp can be written as (ℋ1∪Ω)⊥(\mathcal{H}_1\cup\Omega)^\perp, and ℋ1\mathcal{H}_1 itself can be split as a union ℋ1M∪ℋ1E\mathcal{H}_1^M \cup \mathcal{H}_1^{E}, where the two sub-classes consist of the ℳ\mathcal {M}-arrows and the ℰ\mathcal{E}-arrows of the various h∈ℋh\in\mathcal{H}. The assumptions made on Γ\Gamma and the presence of a generator on A\mathbf{A} entail that ℋ1M\mathcal{H}_1^M is a set, so we can conclude.

The state of the art.

[FK]’s solution of the OSP can be generalized: [AHS] show that ℋ⊥\mathcal{H}^\perp is reflective in a category A\mathbf{A} with a proper FS 𝔽=(ℰ,ℳ)\mathbb{F}=(\mathcal{E},\mathcal{M}) whenever the class ℋ\mathcal{H} is quasi-presentable, namely it can be written as a union ℋ0∪ℋe\mathcal{H}_0\cup \mathcal{H}_e, where ℋe⊂ℰ\mathcal{H}_e\subset \mathcal{E} and ℋ0\mathcal{H}_0 is presentable (in a suitable sense).

The same paper offers a fairly deep point of view about the “weak analogue” of the OSP, which can be regarded as a generalization of the Small Object Argument (SOA) in Homotopical Algebra; if we build the class ℋ□\mathcal{H}^\square of arrows having a non-unique lifting property against each h∈ℋh\in\mathcal{H}, then we can only hope in a weak reflection, where the unit of the adjunction is only weakly universal. In a setting where “things are defined up to homotopy” this can still be enough, provided that we ensure the reflection maps satisfy some additional properties. The additional property requested in the SOA is that the weak reflection maps belong to the cellular closure of ℋ\mathcal{H}, i.e. they can be obtained as a transfinite composition of pushouts of maps in ℋ\mathcal{H}.

The theory of factorization systems is deeply intertwined with the SOA, too: in [SOA] R. Garner defines an “algebraic” Small Object Argument, exploiting a description of OFS and WFS as suitable pairs (comonad,monad)(comonad, monad) over the category AΔ1\mathbf{A}^{\Delta^1}. In this respect I think that the best person which can give us sensible references for this is our boss, since she wrote this paper.

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Re: Categories of Continuous Functors

Bravo! It fills me with joy to see these Kan seminar posts.

Let me add a fact that lots of people here know, but which probably isn’t known widely enough. It’s a simpler way of stating the axioms for an (orthogonal) factorization system, equivalent to Freyd and Kelly’s definition but bypassing the notions of orthogonality/LLP/RLP. Here goes.

Definition A factorization system on a category A\mathbf{A} consists of two subcategories, E\mathbf{E} and M\mathbf{M}, such that:

every isomorphism is in both E\mathbf{E} and M\mathbf{M};

every map in A\mathbf{A} factorizes uniquely up to unique isomorphism as a map in E\mathbf{E} followed by a map in M\mathbf{M}.

I believe this simplification is due to André Joyal. It’s in the appendix of some unpublished notes of his on quasicategories (available as a CRM preprint). But it’s not at all hard to prove.

Whenever gf=1g f=1, and f∈ℰf\in\mathcal{E}, then also g∈ℰg\in\mathcal{E}.

Condition 3 is fairly deep! If we ask “in which factorization systems 𝔽\mathbb{F} the two classes are two-sided 3-for-2?” (notice that every condition admits a dual one on ℳ\mathcal{M}) one is left with something which behaves like a torsion theory in a category: this is what Rosicki and Tholen’s paper is about!

Re: Categories of Continuous Functors

I really like the Joyal definition. It makes it clear that if your factorizations are appropriately “unique”, then they do in fact form an orthogonal factorization system. That is, the definition of an OFS isn’t going to miss any examples for being “too strong”.

Rosicky and Tholen’s condition (1) surprised me, even in the “easy” direction, saying that for any OFS, EE is closed under all colimits in the arrow category. Freyd and Kelly did prove some cocompleteness properties of EE in their Prop 2.1.1, but they didn’t show this.

On the topic of factorization systems, I learned something very basic from Freyd and Kelly: any finitely-complete category admitting arbitrary intersections of monomorphisms has (Strong Epi, Mono) factorizations (Prop 2.3.4). So proper factorization systems abound!

On a more basic level, this says to me that “most categories have enough monos and epis”, because without a factorization system, it seems to me that categorical methods of constructing monos or epis are limited. In fact, categories can be cooked up that have no monos or epis whatsoever. In such a category, for example, it is not very informative to study the subobject lattice of an object. So it’s good to know that under mild-ish completeness conditions, there will be plenty of monos and epis present in the category, and they can be reasonably expected to provide insight into the category’s structure.

Re: Categories of Continuous Functors

Thanks Fosco for this great summary!

I’ve been thinking a bit this morning about the relationship between the construction in Theorem 4.1.3 (solving the orthogonal subcategory problem) and the algebraic small object argument. I’ll confess, I haven’t yet read the AHR paper, so it’s possible that I’m repeated a lot of what is done in there.

Suppose I have a category A\mathbf{A} satisfying the hypotheses of Theorem 4.1.3 and a set of arrows 𝒥\mathcal{J}. There is category 𝒥⊘\mathcal{J}^\oslash (my preferred notation for this has a “□\square” in place of the “O”) with a forgetful functor 𝒥⊘→A2\mathcal{J}^\oslash \to \mathbf{A}^2. Objects are maps in A\mathbf{A} with a chosen solution to any lifting problem against j∈𝒥j \in \mathcal{J}, and morphisms are commutative squares for which the triangle of chosen lifts commutes.

Like ℋ⊥\mathcal{H}^\perp, the category 𝒥⊘\mathcal{J}^\oslash is complete, and 𝒥⊘→A2\mathcal{J}^\oslash \to \mathbf{A}^2 creates these limits. Thus, the forgetful functor admits a left adjoint if and only if it satisfies the solution set condition. Moreover, by a recognition theorem due to John Bourke, if this adjoint exists, then J⊘J^\oslash encodes an algebraic weak factorization system. (You can use Beck’s monadicity theorem to see that such an adjunction is necessarily monadic.)

I claim that the Freyd-Kelly construction can be adapted to produce a solution set, and thus a left adjoint. By uniqueness of adjoints, this must be equivalent to Richard Garner’sconstruction, though I don’t see this directly. (The answer must be in Kelly’s “A unified treatment…” somewhere.)

Here’s how this goes: First note that the arrow category A2\mathbf{A}^2 inherits a proper factorization system, boundedness, and so on from A\mathbf{A}. Given an arrow aa and a commutative square f:a→bf \colon a \to b to an arrow b∈𝒥⊘b \in \mathcal{J}^\oslash, one iteratively constructs subobjects of bb just as in the proof above, except that for the successor step one defines the members of Sα+1S_{\alpha+1} to be quotients of maps

c⊔∐j,Sq(j,c)idcod(j)c \sqcup \coprod_{j, Sq(j,c)} id_{cod(j)}

where c∈Sαc \in S_{\alpha} and j∈𝒥j \in \mathcal{J}.

The solution set is defined to be Sσ∩𝒥⊘S_\sigma \cap \mathcal{J}^\oslash, where σ\sigma is the cardinal bounding both the domains and codomains of 𝒥\mathcal{J}. To see this, first we factor the square f:a→bf \colon a \to b through some aσ∈Sσa_\sigma \in S_\sigma. This is done by a transfinite induction, where a0a_0 is the image of ff, aβa_\beta is the union of the previous stages if β\beta is a limit ordinal, and aα+1a_{\alpha+1} is the image of the morphism

The domain component yy of the square idcod(j)→bid_{cod(j)} \to b is defined to be the chosen solution to the lifting problem j→aα→bj \to a_\alpha \to b.

It remains only to show that aσ∈𝒥⊘a_\sigma \in \mathcal{J}^\oslash. Given a square j→aσj \to a_\sigma, boundedness implies that it factors through some aαa_\alpha. By construction of aα+1a_{\alpha+1}, the map yy factors through the domain of aα+1a_{\alpha+1}, defining a potential solution to the lifting problem j→aσj \to a_\sigma. Because the component of the map aσ→ba_\sigma \to b are monomorphisms, this map is indeed a solution to the desired lifting problem (and by construction the map aσ→ba_\sigma \to b lies in the category 𝒥⊘\mathcal{J}^\oslash).

I suspect that by replacing the coproducts with coends, one could adapt this construction to the case where 𝒥\mathcal{J} is a small category of arrows — the algebraic small object argument works at this level of generality — but I didn’t check this carefully.

Re: Categories of Continuous Functors

Does anyone know what the reference [15, pp. 118-119] is, on p.170 (Introduction) in the paper? There is no [15] in the references. If what is meant was [5], I have no access to Freyd’s Abelian Categories to check, and the pagination is different on the online version. Does anyone know where the results he refers to have been stated/proved elsewhere? Has the general result, the one he claims was asserted in [15, pp. 118-119], been proven?

Related to this, I’m wondering whether there is a more logical perspective one can take on the CFP (and perhaps also the OSP). For instance, take 𝒜=Set\mathcal{A} = Set and 𝒞\mathcal{C} to be a category with enough structure for a particular type of theory 𝕋\mathbb{T} (e.g. Lawvere, regular, cartesian etc.) The models of 𝕋\mathbb{T} are then given by the full subcategory Mod(𝕋)=[𝒞,Set]strucMod(\mathbb{T}) = [\mathcal{C},Set]_{\text{struc}} of [𝒞,Set][\mathcal{C},Set] preserving the relevant structure. For some of those theories we will be able to express [𝒞,Set]struc[\mathcal{C},Set]_{\text{struc}} as Γ(𝒞,Set)\Gamma(\mathcal{C}, Set) where Γ\Gamma is a class of diagrams. For instance, in the case of Lawvere theories, 𝒞𝕋\mathcal{C}_{\mathbb{T}} will be a category with finite products and Γ\Gamma the class of product diagrams in 𝒞𝕋\mathcal{C}_{\mathbb{T}}.

So then we may use the Freyd/Kelly approach to the CFP to see whether the category of models Mod(𝕋)\text{Mod} (\mathbb{T}) of a certain theory is reflective in [𝒞𝕋,Set][\mathcal{C}_{\mathbb{T}} , Set]. Sometimes this is an interesting thing to know, e.g. if you want to find out whether or not the categories of models of particular theories have certain limits. On the other the results indicated by the mysterious reference [15, pp. 118-119] on p.170 have, it seems to me, another interesting logical application, in that they allow you to see when a particular logical condition can be expressed in terms of (limit/colimit) cones. Consider the following argument: Take a theory 𝕋\mathbb{T} (of a particular type) and consider Mod(𝕋)\text{Mod}(\mathbb{T}). If Mod(𝕋)\text{Mod}(\mathbb{T}) is not reflective in [𝒞𝕋,Set][\mathcal{C}_{\mathbb{T}},Set] then it is not of the form Γ(𝒞𝕋,Set)\Gamma(\mathcal{C}_{\mathbb{T}},Set) for some class of cones Γ\Gamma. Therefore its logical structure does not just consists of cone conditions (i.e. it’s not a sketchable?). But perhaps this is not helpful at all since determining whether or not Mod(𝕋)\text{Mod}(\mathbb{T}) is reflective or not in [𝒞𝕋,Set][\mathcal{C}_{\mathbb{T}},Set] would probably require checking if it has or not the limits one is interested in. I don’t really know.

In any case, the connection to the more “logical” side of things seems to be suggested by Freyd and Kelly when they write

Our results seem to bear some relation […] to those of Barr and Schubert on the cocompleteness of the algebras over a monad.

at the end of the Introduction to their paper. So, anyway, I’m wondering if there’s a more logical interpretation of the CFP or OSP.

Re: Categories of Continuous Functors

So, anyway, I’m wondering if there’s a more logical interpretation of the CFP or OSP.

This is an interesting point and I also wondered about this “higher” interpretation.

Studying the paper and some bits of [LPAC] left me with the sensation that lots of interesting points to answer this question are buried in another paper we will see during the seminar, “A Classification of Accessible Categories”. I already knew something about that paper, since I stumbled upon it when I was studying the following situation:

Let 𝒜\mathcal{A} be a class of small categories; for every (small) category C\mathbf{C}, we denote Ind𝒜(C)Ind_{\mathcal{A}}(\mathbf{C}) the completion of C\mathbf{C} by 𝒜\mathcal{A}-shaped diagrams: formally we consider the full subcategory of PSh(C)PSh(\mathbf{C}) closed under colimits indexed by elements of 𝒜\mathcal{A} and containing all the representable presheaves.

Now, let ℬ\mathcal{B} be another class of small categories; denote by 𝕊(ℬ)\mathbb{S}(\mathcal{B}) the sketch in C\mathbf{C} whose colimits are diagrams indexed by a category in ℬ\mathcal{B} (and whose limits are empty).
Now we observed that the relation Ind𝒜(C)=Mod(𝕊(ℬ))Ind_{\mathcal{A}}(\mathbf{C}) = \mathsf{Mod}(\mathbb{S}(\mathcal{B})) holds if and only if the following condition holds:

a small category J\mathbf{J} is in 𝒜\mathcal{A} if and only if J\mathbf{J}-colimits of sets commutes with I\mathbf{I}-limits of sets for every I\mathbf{I} in ℬ\mathcal{B}.

This led me to some interesting conversations with M. Bjerrum, whose thesis projects deals exactly with a similar problem of classification of “which limits commute with which colimits”. But I think this is rather unrelated with your question. I would only let you know that I’m with you in feeling that presentability issues, commutation with certain 𝒜\mathcal{A}-shaped limits, and representations of algebraic theories form the pieces of a huge puzzle I would like to understand.

Re: Categories of Continuous Functors

Dimitris, you inspired me to take a look at Adámek and Rosický’s treatment of limit sketches (starting p. 41 of LPAC). For them, a limit sketch is a small category equipped with a set of small cones; a model is a Set\mathbf{Set}-valued functor which sends the prescribed cones to limit cones.

They show the categories of models of limit sketches are precisely the locally presentable categories. The method basically follows in the footsteps of Freyd and Kelly: they treat the category of models as a small-orthogonality class in a presheaf category and verify the solution set condition via transfinite induction that I assume must be very similar to Freyd and Kelly’s. They’re also able to generalize to models in a locally presentable category other than Set\mathbf{Set} using these methods.

It occurs to me that Freyd and Kelly’s work goes beyond this in that they treat certain large orthogonality classes. But now that I think about this, I’m not sure that very much is gained, because they only treat categories [C,A]Γ[C, A]_{\Gamma} where CC is small. So for example, I don’t think their theory applies to large Lawvere theories like the theory of Compact Hausdorff Spaces. I wonder if the AHS paper Fosco mentioned overcomes this limitation?

The other generalization one might think about in a logical spirit is generalizing from limit-sketches (and locally presentable categories) to limit-colimit-sketches (and accessible categories), but these categories aren’t cocomplete, so they can’t be reflective in a presheaf category, and the theory here doesn’t seem to apply.

Maybe the most significant way the Freyd and Kelly go beyond the theory of sketches is that they can prove reflectivity of [C,A]Γ[C, A]_{\Gamma} for a broader range of AA – for example when A=TopA = \mathbf{Top}, their work shows that the category of topological groups is reflective in [𝕋,Top][\mathbb{T}, \mathbf{Top}] where 𝕋\mathbb{T} is the Lawvere theory of groups. But I guess it depends on your view of logic whether you consider taking models in categories other than Set\mathbf{Set} to be properly “logical”.

Re: Categories of Continuous Functors

I’m not sure that very much is gained, […] I wonder if the AHS paper Fosco mentioned overcomes this limitation?

My sensation is that there’s some subtle set theory hidden behind this (Vopenka principle trivializes every smallness condition in the OSP -but, and this is kinda strange- not in the SOA). Unfortunately I’m not able to say something more since I’m totally ignorant about this (I tried to understand what Vopenka principle is about, but I failed…).

Re: Categories of Continuous Functors

They show the categories of models of limit sketches are precisely the locally presentable categories. The method basically follows in the footsteps of Freyd and Kelly: they treat the category of models as a small-orthogonality class in a presheaf category and verify the solution set condition via transfinite induction that I assume must be very similar to Freyd and Kelly’s.

Indeed, this is the case that I had in the back of my head when I thought about the logical aspects of the CFP. It is a very nice result because it essentially provides a purely categorical characterization of those categories that are categories of models of algebraic theories (if by “algebraic theories” we understand ”limit sketches”.)

[…] to limit-colimit-sketches (and accessible categories), but these categories aren’t cocomplete, so they can’t be reflective in a presheaf category, and the theory here doesn’t seem to apply.

The fact that a characterization of categories of models of theories as reflective subcategories of functor categories fails when these theories give accessible model categories, raises the question whether there are any other type of theories whose model categories can be characterized as reflective subcategories of functor categories. Since we know that the locally presentable categories are exactly the (co)complete accessible ones and accessible categories can be characterized as those categories that are categories of models of some theory, then this won’t be the case for any non-algebraic theory. However this does not rule out an ”exotic” type of theory, lying outside the standard system of classification.

In any case, the cocompleteness constraint you mention is very severe, so I doubt whether there would be such strange, exotic theories. Perhaps something of relevance is in the logical sections of the [AHS] paper.

Re: Categories of Continuous Functors

I think the appropriate “no-go” theorem here is a result of Rosicky, Trnkova, and Adamek saying that under Vopenka’s principle, any cocomplete category with with a small dense generator is locally presentable. (And the converse holds – the theorem is equivalent to Vopenka). Confusingly, Adamek and Rosicky use the term “bounded” to refer to a category with a small, dense generator!

If I’m not mistaken, if CC is a small category, then any full subcategory of [C,Set][C, \mathbf{Set}] containing the representables has those representables as a small, dense subcategory. So if we want to go beyond the locally presentable case, then something about the whole setup needs to change radically.

I find it interesting just to learn these equivalent statements of Vopenka’s principle, even if it still remains a black box that I don’t understand!

Re: Categories of Continuous Functors

If 𝒞\mathcal{C} is a category with dense full subcategory 𝒢\mathcal{G} and 𝒟\mathcal{D} is a full subcategory with 𝒢⊆𝒟⊆𝒞\mathcal{G} \subseteq \mathcal{D} \subseteq \mathcal{C}, then 𝒟\mathcal{D} is also dense. The proof is not too hard, but I haven’t been able to figure out the “real reason” why this is true.

Re: Categories of Continuous Functors

Here’s the reason that makes sense to me: G⊆CG \subseteq C is dense just if the restricted Yoneda embedding C→[Gop,Set]C \to [G^{\mathrm{op}}, \mathbf{Set}], c↦Hom(−,c)c \mapsto \mathrm{Hom}(-, c) is fully faithful. Then if you restrict a fully faithful functor to a full subcategory, it’s still fully faithful.

Re: Categories of Continuous Functors

Here is a proof in the language of proarrow equipments, which shows that it does hold in the enriched case, and also that it is “formal” in some sense. In an equipment every arrow f:A→Bf:A\to B induces an adjoint pair of proarrows f•:A⇸Bf_\bullet : A &#x21F8; B and f•:B⇸Af^\bullet: B &#x21F8; A with f•⊣f•f_\bullet \dashv f^\bullet. We say ff is fully faithful if the unit 1A→f•⊙f•1_A \to f_\bullet \odot f^\bullet is an isomorphism (I write composition of proarrows in diagrammatic order with ⊙\odot). We say g:B→Cg:B\to C is the (pointwise) left Kan extension of h:A→Ch:A\to C along ff, written g=Lanhfg = Lan_h f, if we have an isomorphism g•≅(h•⊳f•)g^\bullet \cong (h^\bullet \rhd f^\bullet), where ⊳\rhd is the right hom adjoint to ⊙\odot. Finally, we say f:A→Bf:A\to B is dense if the obvious map UB→(f•⊳f•)U_B \to (f^\bullet \rhd f^\bullet) is an isomorphism; according to the previous sentence this says that 1B=Lanff1_B = Lan_f f, while interpreted for profunctors it says that the restricted Yoneda embedding B→PAB\to P A is fully faithful.

Re: Categories of Continuous Functors

Mike, thanks for this. I haven’t digested this formalism yet but on first inspection I’m a little confused as to how the symbols ⊙\odot and ⊳\rhd interact and especially by how I am to understand ⊳\rhd. Initially I took ⊳\rhd to be the internal hom, but equations like (g•⊙(gf)•)⊳(gf)•=g•⊳((gf)•⊳(gf)•)(g_\bullet \odot (gf)^\bullet) \rhd ( gf )^\bullet = g_\bullet \rhd (( gf )^\bullet \rhd ( gf )^\bullet)
suggest (to me) that ⊳\rhd also stands for an arrow in the ambient category, in this case the bicategory of proarrows. Is this just the standard practice of interchanging internal with external homs in closed monoidal categories, since the latter can be recovered from the former? Or is something else going on, i.e. is there some rule of the ⊙\odot, ⊳\rhd symbolism that you are using that is supposed to encode the aforementioned interchangeability? (I’m basically confused as to how much of this argument is formal/axiomatic in the setting of proarrow equipments and how much of it uses standard properties of (closed) monoidal categories - or perhaps there is no distinction between the two?)

Re: Categories of Continuous Functors

⊳\rhd is the right bicategorical internal hom in the closed bicategory of proarrows. This means that for proarrows M:A→BM: A \to B, N:B→CN: B \to C and P:A→CP:A\to C we have natural adjunction isomorphisms

Re: Categories of Continuous Functors

This is nifty – and it motivates learning the calculational rules of an equipment! I suppose the crux of the matter is the equation g=Lanf(gf)g = \mathrm{Lan}_f(gf): this is a fact whose significance probably bears some mulling over.

I notice, Mike, that you write equality for isomorphism, and I suspect you have good reason. The question that occurs to me is: how do you know that the isomorphisms you end up with are the “right” ones: for example, that the isomorphism f•⊳f•≅1Bf^{\bullet} \rhd f^{\bullet} \cong 1_B is “the obvious one”? Is this just a matter of the usual level of informality when a category theorist writes a chain of isomorphisms, or is there a coherence theorem behind the scenes?

Re: Categories of Continuous Functors

In general, yes, you would have to check that the isomorphism constructed is the desired map. However, in this case there’s a sneaky trick which saves you the work. Namely, f•⊳f•f^\bullet \rhd f^\bullet is a monad in the bicategory of proarrows, hence a monoid in some monoidal category, and the obvious map 1→f•⊳f•1\to f^\bullet \rhd f^\bullet is its unit transformation. And it’s a general fact that if a monoid MM in any monoidal category is isomorphic to the unit object II by any old random isomorphism (not necessarily having anything to do with the monoid structure), then in fact the unit transformation I→MI\to M is an isomorphism. This is a nice exercise.

Re: Categories of Continuous Functors

I think the terminology I was using expressed essentially the same principle, though I was not aware of the connection with Vopenka’s principle, of which I don’t know what to make. It’s very interesting though, and probably worth looking into deeper.

I linked the page about local objects just to point out clearly that “k⊥Bk\perp B” is only the ancient name for “BB is {k}\{k\}-local”. I think that they are called “local” objects for their link to (GZ-)localizations (my sensation is that an echo of this is in the theory of Bousfield localizations), since one can see pretty easily that if I localize C\mathbf{C} with respect to (the saturation of) ℋ\mathcal{H}, then C[ℋ−1]≅ℋ⊥\mathbf{C}[\mathcal{H}^{-1}]\cong \mathcal{H}^\perp.

I’m aware I’m being too much sketchy, I hope to return on this during the next days… But somebody else will certainly explain it much better. Sorry!

Re: Categories of Continuous Functors

I wonder what more can be said about the class of categories that Freyd and Kelly choose to work with. The list of conditions is quite long: cocomplete, equipped with a proper factorization system (E,M)(E, M), MM-bounded and possessing an EE-generator, and EE-co-well-powered. It turns out that in Basic Concepts of Enriched Category Theory, Kelly calls essentially this notion a locally bounded category, except that EE-co-well-poweredness is weakened to admitting arbitrary EE-cointersections.

With so much structure, what else comes along for the ride? According to the nlab page, any cocomplete and EE-cocomplete (this seems to mean admitting arbitrary EE-cointersections) category with an EE-generator is total, so in particular completeness comes for free. Freyd and Kelly’s Cor. 2.5.2 then yields MM-well-poweredness. This is without even using MM-boundedness.

I wonder what other nice properties follow from the hypotheses Freyd and Kelly use?